BigDecimal(Variable Precision Floating Library for Ruby)

Japanese

BigDecimal is an extension library for the Ruby interpreter. Using BigDecimal class, you can obtain any number of significant digits in computation. For the details about Ruby see:

http://www.ruby-lang.org/en/:Official Ruby page(English).
http://ruby.freak.ne.jp/:Ruby informations(Japanese).
http://kahori.com/ruby/ring/:Mutually linked pages relating to Ruby(Japanese).

NOTE:
This software is provided "AS IS" and without any express or implied warranties,including,without limitation,the implied warranties of merchantibility and fitness for a particular purpose. For the details,see COPYING and README included in this distribution.

Introduction

Usage and methods
Infinity,NaN,Zero
Internal structure
Binary or decimal number representation
Resulting number of significant digits

Introduction

Ruby already has builtin (variable length integer number) class Bignum. Using Bignum class,you can obtain any integer value in magnitude. But, variable length floating number class is not yet built in. This is why I made variable length floating class BigDecimal. Feel free to send any comments or bug reports to me. shigeo@tinyforest.gr.jp I will try(but can't promise) to fix bugs reported.

Installation

The Ruby latest version can be downloaded from Official Ruby page. Once decompress the downloaded Ruby archive,follow the normal installation procedures according to the documents included.

Usage and methods

Suppose you already know Ruby programming, to create BigDecimal objects,the program would like:

   require 'bigdecimal'
   a=BigDecimal::new("0.123456789123456789")
   b=BigDecimal::new("123456.78912345678",40)
   c=a+b

List of methods

In the following explanations,n specifies the minimum number of resulting significant digits, not exactly but slightly excess memories will be allocated to newly created object. In 32 bits integer system,every 4 digits(in decimal) are computed simultaneously. This means the number of significant digits in BigDecimal is always a multiple of 4.

new

"new" method creates a new BigDecimal object.
a=BigDecimal::new(s[,n])
where:
s: Initial value string.
n: Maximum number of significant digits of a. n must be a Fixnum object. If n is omitted or is equal to 0,then the maximum number of significant digits of a is determined from the length of s.

double_fig

double_fig is a class method which returns the number of digits the Float class can have.

  p BigDecimal::double_fig  # ==> 20 (depends on the CPU etc.)

The equivalent C programs which calculates the value of double_fig is:

 double v          = 1.0;
 int    double_fig = 0;
 while(v + 1.0 > 1.0) {
    ++double_fig;
    v /= 10;
 }

prec

r,m = a.prec
where r is the number of significant digits of a, m is the maximum number of significant digits a can hold.

  require "bigdecimal"
  a = BigDecimal.new("0.12345")
  p a.prec                              # ==> [8, 12]
  b = BigDecimal.new("0.1234500000000")
  p b.prec                              # ==> [8, 20]
  c = BigDecimal.new("0.12345",20)
  p c.prec                              # ==> [8, 24]

r and m are always the multiple of log10(BigDecimal::BASE).

addition(c = a + b)
For the resulting number of significant digits of c,see

Resulting number of significant digits

assign

add

sub

mult

div

fix

frac

floor[(n)]

ceil[(n)]

round[(n)]

truncate[(n)]

divmod

remainder

abs

to_i

to_s[(n)]

exponent

to_f

BASE

mode

limit[(n)]

sign

Internal structure

nan?

infinite?

finite?

to_parts

inspect

dup

sqrt

sincos

exp

power

zero?

nonzero?

<=>

Following methods need no explanation.

About 'coerce'

For the binary operation like A op B:

1.Both A and B are BigDecimal objects: A op B is normally performed.
2.A is the BigDecimal object but B is other than BigDecimal object: Operation is performed,after B is translated to correcponding BigDecimal object(because BigDecimal supports coerce method).
3.A is not the BigDecimal object but B is BigDecimal object: If A has coerce mthod,then B will translate A to corresponding BigDecimal object and the operation is performed,otherwise an error occures.

Attention must be paid when a String is to be translated to BigDecimal. Translation stops without error at the character representing non digit. For instance,"10XX" is translated to 10,"XXXX" is translated to 0.
String representing zero or infinity such as "Infinity","+Infinity","-Infinity",and "NaN" can also be translated to BigDecimal unless false is specified by mode method.
BigDecimal class supports coerce method(for the details about coerce method,see Ruby documentations). This means the most binary operation can be performed if the BigDecimal object is at the left hand side of the operation.

For example:

  a = BigDecimal.E(20)
  c = a * "0.123456789123456789123456789" # A String is changed to BigDecimal object.

is performed normally.
But,because String does not have coerce method,the following example can not be performed.

  a = BigDecimal.E(20)
  c = "0.123456789123456789123456789" * a # ERROR

If you actually have any inconvenience about the error above. You can define a new class derived from String class, and define coerce method within the new class.

Infinity,Not a Number(NaN),Zero

Infinite numbers and NaN can be represented by string writing "+Infinity"(or "Infinity"),"-Infinity",and "NaN" respectively in your program. Infinite numbers can be obtained by 1.0/0.0(=Infinity) or -1.0/0.0(=-Infinity).

NaN(Not a number) can be obtained by undefined computation like 0.0/0.0 or Infinity-Infinity. Any computation including NaN results to NaN. Comparisons with NaN never become true,including comparison with NaN itself.

Zero has two different variations as +0.0 and -0.0. But,still, +0.0==-0.0 is true.

Computation results including Infinity,NaN,+0.0 or -0.0 become complicated. Run following program and comfirm the results. Send me any incorrect result if you find.


 require "bigdecimal"
 aa  = %w(1 -1 +0.0 -0.0 +Infinity -Infinity NaN)
 ba  = %w(1 -1 +0.0 -0.0 +Infinity -Infinity NaN)
 opa = %w(+ - * / <=> > >=  < == != <=)
 for a in aa
  for b in ba
    for op in opa
      x = BigDecimal::new(a)
      y = BigDecimal::new(b)
      eval("ans= x #{op} y;print a,' ',op,' ',b,' ==> ',ans.to_s,\"\n\"")
    end
  end
 end

Internal structure

BigDecimal number is defined by the structure Real in BigDecimal.h. Digits representing a float number are kept in the array frac[] defined in the structure. In the program,any floating number(BigDecimal number) is represented as:
= 0.xxxxxxxxx*BASE**n

where 'x' is any digit representing mantissa(kept in the array frac[]), BASE is base value(=10000 in 32 bit integer system), and n is the exponent value.
Larger BASE value enables smaller size of the array frac[],and increases computation speed. The value of BASE is defined ind VpInit(). In 32 bit integer system,this value is 10000. In 64 bit integer system,the value becomes larger. BigDecimal has not yet been compiled and tested on 64 bit integer system. It will be very nice if anyone try to run BigDecimal on 64 bit system and inform me the results. When BASE is 10000,an element of the array frac[] can have vale of from 0 to 9999. (up to 4 digits).
The structure Real is defined in bigdecimal.h as:

  typedef struct {
     VALUE  obj;     /* Back pointer(VALUE) for Ruby object.         */
     unsigned long MaxPrec; /* The size of the array frac[]          */
     unsigned long Prec;    /* Current size of frac[] actually used. */
     short    sign;         /* Attribute of the value.  */
                            /*  ==0 : NaN               */
                            /*    1 : +0                */
                            /*   -1 : -0                */
                            /*    2 : Positive number   */
                            /*   -2 : Negative number   */
                            /*    3 : +Infinity         */
                            /*   -3 : -Infinity         */
     unsigned short flag;   /* Control flag             */
     int      exponent;     /* Exponent value(0.xxxx*BASE**exponent) */
     unsigned long frac[1]; /* An araay holding mantissa(Variable)   */
  } Real;

The decimal value 1234.56784321 is represented as(BASE=10000):

    0.1234 5678 4321*(10000)**1

where frac[0]=1234,frac[1]=5678,frac[2]=4321, Prec=3,sign=2,exponent=1. MaxPrec can be any value greater than or equal to Prec.

Binary or decimal number representation

I adopted decimal number representation for BigDecimal implementation. Of cource,binary number representation is common on the most computers.

Advantages using decimal representation

The reason why I adopted decimal number representation for BigDecimal is:

Easy for debugging

The floating number 1234.56784321 can be easily represented as:
frac[0]=1234,frac[1]=5678,frac[2]=4321,exponent=1,and sign=2.

Exact representation

Following program can add all numbers(in decimal) in a file without any error(no round operation).


   file = File::open(....,"r")
   s = BigDecimal::new("0")
   while line = file.gets
      s = s + line
   end

If the internal representation is binary,translation from decimal to binary is required and the translation error is inevitable. For example, 0.1 can not exactly be represented in binary.
0.1 => b1*2**(-1)+b1*2**(-2)+b3*2**(-3)+b4*2**(-4)....
where b1=0,b2=0,b3=0,b4=1...
bn(n=1,2,3,...) is infinite series of digit with value of 0 or 1, and rounding operation is necessary but where we should round the series ? Of cource,exact "0.1" is printed if the rouding operation is properly done,

Significant digit we can have is automatically determined

In binary representation,0.1 can not be represented in finite series of digit. But we only need one element(frac[0]=1) in decimal representation. This means that we can always determine the size of the array frac[] in Real structure.

Disadvantage of decimal representation

Advantages stated so far can also be disadvantages if the input from outside is represented in binary. Translation error from decimal to binary or vice versa is inevitable. So,translation from Float(binary) to BigDecimal(decimal) is not alway done exactly.

Which is the first input?

Because most people uses decimal notatin for numeric data representation, BigDecimal can handle numeric data without loss of translation error.

Resulting number of significant digits

For the fundamental arithmetics such as addition,subtraction, multiplication,and division,I prepared 2 group of methods

1. +,-,*,/

For the operation + - * /,you can not specify the resulting number of significant digits.
Resulting number of significant digits are defined as:
1.1 For * and /,resulting number of significant digits is the sum of the significant digits of both side of the operator.
1.2 For + and -,resulting number of significant digits is determined so that no round operation is needed.
For example, c has more than 100 siginificant digits if c is computed as:
c = 0.1+0.1*10**(-100)

As +,-,and * are always exact(no round operation is performed), which means more momories are required to keep computation results. As for the division as c = a/b,the significant digits of c is the same as a*b. Division such as c=1.0/3.0 will be rounded.

2. assign,add,sub,mult,div

The length of the significant digits obtained from +,-,*,/ is always defined by that of right and left side of the operator. To specify the length of the significant digits by your self, use methos assign,add,sub,mult,div, or limit(class method). Following example compute the ratio of the circumference of a circle to its dirmeter(pi=3.14159265358979....) using J.Machin's formula.

#!/usr/local/bin/ruby

#
# pi.rb
#  USAGE: ruby pi.rb n
#   where n is the number of digits required.
#  EX.: ruby pi.rb 1000
#

require "bigdecimal"
#
# Calculates 3.1415.... using J. Machin's formula.
#
def big_pi(sig) # sig: Number of significant figures
  exp    = -sig
  pi     = BigDecimal::new("0")
  two    = BigDecimal::new("2")
  m25    = BigDecimal::new("-0.04")
  m57121 = BigDecimal::new("-57121")

  u = BigDecimal::new("1")
  k = BigDecimal::new("1")
  w = BigDecimal::new("1")
  t = BigDecimal::new("-80")
  while (u.exponent >= exp) 
    t   = t*m25
    u,r = t.div(k,sig)
    pi  = pi + u
    k   = k+two
  end

  u = BigDecimal::new("1")
  k = BigDecimal::new("1")
  w = BigDecimal::new("1")
  t = BigDecimal::new("956")
  while (u.exponent >= exp )
    t,r = t.div(m57121,sig)
    u,r = t.div(k,sig)
    pi  = pi + u
    k   = k+two
  end
  pi
end

if $0 == __FILE__
  print "PI("+ARGV[0]+"):\n"
  p pi(ARGV[0].to_i)
end

Shigeo Kobayashi (E-Mail:<shigeo@tinyforest.gr.jp>)

BigDecimal(Variable Precision Floating Library for Ruby)

Contents

Introduction

Installation

Usage and methods

List of methods

About 'coerce'

Infinity,Not a Number(NaN),Zero

Internal structure

Binary or decimal number representation

Advantages using decimal representation

Disadvantage of decimal representation

Which is the first input?

Resulting number of significant digits

1. +,-,*,/

2. assign,add,sub,mult,div